Dic₁₀₅ #88286

order	420
isoclinism	isoclinic class of D₁₀₅
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₀₅
unambiguous semidirect product	C₃⋊Dic₃₅
unambiguous semidirect product	C₇⋊Dic₁₅

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₀₅
commutator, hypocenter	C₃×C₅×C₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇	C₂
LCS	G ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 420

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19);(1,3,2,4)(6,7)(9,12)(10,11)(14,19)(15,18)(16,17)

Polycyclic presentation:

2^2,3,5,7|a^2=b,c^a=c^2,d^a=d^4,e^a=e^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 11);